1. Core 4 OCR How does the day go?
Session 1 – Algebra
Session 2 – Differentiation and Integration 1
Session 3 – Integration 2 and differential equations
Session 4 – Vectors If you would like copies of the Powerpoints please visit the Resources page on my website

2. Algebra - What you need to know
How to simplify algebraic fractions
How to divide a polynomial by a linear or quadratic expression
How to write a fraction as the sum of partial fractions
How to expand a bracket using the Binomial Expansion for any power
How to write a curve given parametrically as the Cartesian equation.

3. Algebraic division – boxes method
You may be good at this – keep doing the way you know
If this is causing you problems, this tabular method might help you

4. Working with cubics
Write the x -2 on the outside and the first term on the inside x
2x3 -2
Fill in the first column

5. Working with cubics
How many more x2 do we need?
2x2 x
2x3 -2
-4x2

6. Working with cubics
Finish off the table writing an extra number in a box to the right for the remainder
2x2 x
2x3
7x2 -2
-4x2

7. Working with cubics 2x2 7x +7 x 2x3 7x2 -2 -4x2 -14x -14 18
Quotient is 2x2 +7x +7 and remainder 18
WARNING: DON’T use this method if the question says “use the remainder theorem” to find the remainder. You’ll get NO MARKS!

8. 2.Simplifying algebraic fractions Cancel factors in the top with factors in the bottom
(June 08 q1) (a) Simplify

9. 2b Dividing by quadratics Divide by .x x2 x3 4x +1

10. 2b Dividing by quadratics Divide by .x x2 x3 4x
4x2 +1

11. Dividing by quadratics Divide by .x x2 x3
-2x2 4x
4x2 +1

12. 2b Dividing by quadratics Divide by .x -2 x2 x3
-2x2 4x
4x2 +1

13. Dividing by quadratics Divide by .x -2 x2 x3
-2x2 4x
4x2
-8x +1

14. 2b Dividing by quadratics Divide by .x -2 x2 x3
-2x2 4x
4x2
-8x +1 x
- 3
Quotient is x – 2 and remainder x – 3

15. 3. A trickier example
When is divided by the quotient is and the remainder is Find the values of the constants a, b and c

16. Rewriting in terms of multiplication Constant term (or substituting x=0) Cubic term Linear term Quadratic term can be used as a check

17. Partial fractions
Look at the denominator (bottom) to know how to begin
Partial fractions for distinct linear factors (could be two or three)
Partial fractions with a repeated factor

18. Distinct linear factors
4 Express x in partial fractions
( x – 2 ) ( x - 3 )
x = A + B
( x – 2 ) ( x - 3 ) ( x – 2 ) (x - 3 )
Multiply by denominator…
x = A ( x – 3 ) + B (x – 2)
Substitute x = 3 to ‘remove’ A = B
Substitute x = 2 to ‘remove’ B = -A, A = -2
x =
( x – 2 ) ( x - 3 ) ( x – 2 ) (x - 3 )

19. 5 Repeated factor x – 1 = A + B + C
( x + 1 ) ( x - 2 ) ( x + 1 ) (x - 2 ) (x - 2 )2
Multiply by denominator
x – = A ( x - 2 ) B ( x + 1 )( x - 2 ) C ( x + 1 )
Substitute x = -1
= A B C 0
A = /9
Substitute x = 2
C = 1/3

20. x – 1 = A ( x - 2 )2 + B ( x + 1 )( x - 2 ) + C ( x + 1 )
Equate the x2 coefficients
0 = A B
B = 2/9
x – = / / / 3
( x + 1 ) ( x - 2 ) ( x + 1 ) (x - 2 ) (x - 2 )2
Now rewrite
x – =
( x + 1 ) ( x - 2 ) ( x + 1 ) (x - 2 ) (x - 2 )2

21. Binomial expansion How to expand (1 + x ) to a power
Positive whole number for C1
Negative or fraction power for C4
The expansion is infinite
and only valid for |x|<1

22. Example 6
The expansion is infinite and only valid for |x|<1

23. Approximation
We use the first part of the expansion as an approximation for small values of x
This could be called a quadratic approximation – but how good is it?

24. Look at the graphs

25. Look at quadratic and cubic approximations

26. What happens if the number is not 1? Example 7

27. Example 6
Find a quadratic approximation for

28. Putting it all together – Example 8

30. Using graphs to check

31. Parametric equations
A curve is given parametrically when both x and y are given in terms of a third variable, often t or θ

32. Eliminating the parameter Example 10
Find the Cartesian equation of the curve given parametrically by

33. A tricky one – Example 11
Find the Cartesian equation of the curve given by